Find three positive numbers in the ratio $2 : 3 : 5,$ the sum of whose squares is $950$

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Solution

It is given that
Ratio of three positive numbers $= 2 : 3 : 5$
Sum of their squares $= 950$
Consider
First number $2 x$
Second number $3 x$
Theird number $= 5 x$
It can be written as
$(2 x)^{2} + (3 x)^{2} + (5 x)^{2} = 950$
By further calculation
$4 x^{2} + 9 x^{2} + 25 x^{2} = 950$
$38 x^{2} = 950$
SO we get
$X^{2} = 950 / 38 = 25$
$x = \sqrt{25} = 5$
Here
First number $= 2 \times 5 = 10$
Second number $= 3 \times 5 = 15$
Third number $= 5 \times 5 = 25$

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